Field of the Invention
The invention relates to a method and a device for cryptographic processing with the aid of an elliptic curve on a computer.
A finite body is called a finite field. Reference may be made to Lidl and Niederreiter: Introduction to Finite Fields and Their Applications, Cambridge University Press, Cambridge 1986, ISBN 0-521-30706-6, p. 15, 45, concerning the properties and definition of the finite field.
Increasingly growing demands are being placed on data security with the wide dissemination of computer networks and associated applications which are being developed over electronic communication systems (communications networks). The aspect of data security takes account of, inter alia,                the possibility of a failure of data transmission;        the possibility of corrupted data;        the authenticity of the data, that is to say the possibility of establishing, and the identification of a sender; and        the protection of the secrecy of the data.        
A “key” is understood as data which are used in cryptographic processing. It is known from public-key methods to use a secret and a public key. Reference is had, in this context, to Christoph Ruland: Informationssicherheit in Datennetzen [Information Security in Data Networks], DATACOM-Verlag, Bergheim 1993, ISBN 3-892238-081-3, p. 73–85.
An “attacker” is defined as an unauthorized person who aims at obtaining the key or breaking the key.
Particularly in a computer network, but increasingly also in portable media, for example a mobile telephone, a chip card or smart card, it is to be ensured that a stored key also cannot be accessed when an attacker takes over the computer, the mobile telephone or the chip card.
In order to ensure adequate security of cryptographic methods, keys, in particular in the case of asymmetric methods, are respectively determined with lengths of several 100 bits. A memory area of a computer or portable medium is mostly of meager dimension. A length of a key of several 100 bits stored in such a memory area reduces the free memory space on the computer or the medium, such that only a few such keys can be stored at the same time.
An elliptic curve and its use in cryptographic processing are known in the literature, for example: Neal Koblitz: A Course in Number Theory and Cryptography, Springer Verlag, New York, 1987, ISBN 0-387-96576-9, p. 150–79; and Alfred J. Menezes: Elliptic Curve Public Key Cryptosystems, Luwer Academic Publishers, Massachusetts 1993, ISBN 0-7923-9368-6, p. 83–116.